Question

# A spring hanging vertically under its own weight has length 20 cm. When a 6.5 kg...

A spring hanging vertically under its own weight has length 20 cm. When a 6.5 kg mass is attached to the end of it, it stretches vertically so that its new length is 117 cm when hanging at rest. If we remove the 5.8 kg mass, attach a 8.5 kg mass to the end of the spring, and again allow it to hang vertically, how long will the spring be?

Original length of the spring, L = 20.0 cm = 0.20 m

When a mass of 6.5 kg is attached to it, new length of the spring, L1 = 117 cm = 1.17 m

So, expansion in the spring, L = L1 - L = 1.17 - 0.20 = 0.97 m

Suppose, k = spring constant of the spring.

So, we have the following equation -

k*L = m*g

=> k*0.97 = 6.5*g--------------------------------------------------------(i)

Now, we remove 5.8 kg and attach 8.5 kg to the spring.

So, the total mass attached to the spring, m' = 6.5 - 5.8 + 8.5 = 9.2 kg

Suppose expansion in the spring in this case = L'

So we have -

k*L' = 9.2*g --------------------------------------------------------(ii)

From equation (i) and (ii) -

k*L' / k*0.97 = 9.2*g / 6.5*g

=> L' = (9.2 x 0.97) / 6.5 = 1.373 m = 137.3 cm

Hence, new length of the spring = 20 + 137.3 = 157.3 cm (Answer)